| L(s) = 1 | + 2-s + 1.21·3-s + 4-s − 2.48·5-s + 1.21·6-s − 0.502·7-s + 8-s − 1.52·9-s − 2.48·10-s + 3.06·11-s + 1.21·12-s − 3.30·13-s − 0.502·14-s − 3.01·15-s + 16-s − 1.81·17-s − 1.52·18-s + 2.40·19-s − 2.48·20-s − 0.609·21-s + 3.06·22-s − 1.89·23-s + 1.21·24-s + 1.18·25-s − 3.30·26-s − 5.49·27-s − 0.502·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.700·3-s + 0.5·4-s − 1.11·5-s + 0.495·6-s − 0.189·7-s + 0.353·8-s − 0.509·9-s − 0.786·10-s + 0.923·11-s + 0.350·12-s − 0.917·13-s − 0.134·14-s − 0.778·15-s + 0.250·16-s − 0.440·17-s − 0.360·18-s + 0.550·19-s − 0.555·20-s − 0.132·21-s + 0.652·22-s − 0.394·23-s + 0.247·24-s + 0.236·25-s − 0.648·26-s − 1.05·27-s − 0.0948·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 1.21T + 3T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 + 0.502T + 7T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 19 | \( 1 - 2.40T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 0.190T + 31T^{2} \) |
| 37 | \( 1 - 6.66T + 37T^{2} \) |
| 41 | \( 1 + 9.80T + 41T^{2} \) |
| 43 | \( 1 + 9.18T + 43T^{2} \) |
| 47 | \( 1 + 3.96T + 47T^{2} \) |
| 53 | \( 1 + 7.02T + 53T^{2} \) |
| 59 | \( 1 - 6.42T + 59T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 + 6.53T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 1.74T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 2.34T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.186037984134229880978060404544, −7.28214474488004730125837038382, −6.71573580618819199800680379608, −5.84474419348498407924823678317, −4.80305201560256106055739772658, −4.22187222588353824193098315595, −3.33699896643441841249364015821, −2.87762918760964245420101670058, −1.68214573809497490088961227535, 0,
1.68214573809497490088961227535, 2.87762918760964245420101670058, 3.33699896643441841249364015821, 4.22187222588353824193098315595, 4.80305201560256106055739772658, 5.84474419348498407924823678317, 6.71573580618819199800680379608, 7.28214474488004730125837038382, 8.186037984134229880978060404544
